Unit 2 · Topic 2.6 Beta

Gravitational Force

Every two masses attract with |F_g| = Gm₁m₂/r², measured center to center; the field g = F/m is about 9.8 N/kg at Earth's surface.

Practice 2: Mathematical RoutinesPractice 3: Scientific Questioning and Argumentation

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Question set for this topic

Part 1 · Hook

Why this matters

The force that pulls an apple off a tree also holds the Moon in its orbit, and it weakens in a precise way as you move away. Stand on a bathroom scale in an elevator, though, and the number changes as you start and stop, even though Earth pulls on you just as hard. This topic separates the pull of gravity from what you actually feel.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Near Earth, the weight of a 2.0 kg book is about

  1. 20 N
  2. 2.0 N
  3. 2.0 kg
  4. 0.20 N
Show the answer

F_g = mg = 2.0 kg × 9.8 m/s² ≈ 20 N.

  • Correct: 20 N:
  • 2.0 N:
  • 2.0 kg:
  • 0.20 N:

2. A 60 kg person in an elevator accelerates upward at 1.0 m/s². The net force on her is

  1. 60 N up
  2. 588 N up
  3. zero
  4. 60 N down
Show the answer

ΣF = ma = 60 kg × 1.0 m/s² = 60 N, along the acceleration.

  • Correct: 60 N up:
  • 588 N up:
  • zero:
  • 60 N down:

3. If a quantity is proportional to 1/r², tripling r

  1. divides it by 9
  2. divides it by 3
  3. multiplies it by 9
  4. leaves it unchanged
Show the answer

1/3² = 1/9.

  • Correct: divides it by 9:
  • divides it by 3:
  • multiplies it by 9:
  • leaves it unchanged:

Part 4 · See it

See it first

A 70 kg rider on a scale in an elevator, three times, with g = 9.8 m/s². Accelerating up at 1.2 m/s²: F_N, scale on rider, is longer than F_g, Earth on rider, and the scale reads 770 N. Constant velocity: equal arrows, 686 N. Accelerating down at 1.2 m/s²: F_N is shorter than F_g, 602 N. The gravitational force, 686 N, is the same in all three.
The scale reads the normal force. Earth's pull, mg, is the same in all three. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Two objects have masseach pulls the other with a gravitational force Gm₁m₂/r², toward the other's center
  2. You divide that force by the mass it acts onyou get the gravitational field g = F/m, which near Earth's surface is about 9.8 N/kg and nearly constant
  3. An object stands on a scale and accelerates up or downthe normal force differs from mg: F_N = m(g + a) going up, m(g − a) going down
  4. The object and scale fall freely togetherthe normal force is zero, so the object is weightless even though gravity still acts

Part 6 · Key ideas

Key ideas

  • Newton's law of universal gravitation: |Fg| = Gm₁m₂/r², attractive, along the line between centers, with r from center to center and G = 6.67 × 10⁻¹¹ N·m²/kg².
  • The gravitational field g = Fg/m = GM/R² at a planet's surface. Near Earth g ≈ 9.8 N/kg = 9.8 m/s², treated as constant.
  • Apparent weight is the normal force (what a scale reads). It equals mg only when the acceleration is zero; in free fall it is zero (weightless).
  • Inertial mass (resisting acceleration) and gravitational mass (feeling gravity) are equal by experiment, so all objects fall with the same g.

Part 7 · Misconception

A common mistake

The wrong idea: Astronauts in orbit are weightless because there is no gravity in space.

What actually happens: At the station's height gravity is about 90% as strong as on the ground. The astronauts and the station fall together around Earth, so the floor pushes on them with zero force: zero apparent weight.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Data table

Scale readings during an elevator ride

A 60 kg student stands on a bathroom scale in an elevator that starts at rest on the ground floor and stops at rest on an upper floor. She records the scale reading at several times. Use g = 9.8 m/s².

Scale reading during the ride
Time (s)Scale reading (N)
0.0588
2.0660
4.0588
8.0588
10.0516
12.0588

1. At 2.0 s the scale reads 660 N. What is the elevator doing then?

  1. Speeding up while moving upward
  2. Moving upward at a constant speed
  3. Slowing down while moving upward
  4. At rest, waiting at the ground floor
Show the answer

The scale pushes up on the student with 660 N while Earth pulls her down with mg = 588 N. The net force is 72 N up, so she accelerates upward. Starting from rest, that means speeding up on the way up.

  • Correct: Speeding up while moving upward: Right: the normal force exceeds the weight, so the acceleration points up.
  • Moving upward at a constant speed: At constant speed the scale would read her weight, 588 N.
  • Slowing down while moving upward: Slowing down on the way up needs a downward acceleration, so a reading below 588 N.
  • At rest, waiting at the ground floor: At rest the scale reads 588 N, her weight.

2. What is the size of the elevator's acceleration when the scale reads 660 N, in m/s²?

Type a number in m/s^2.

Show the answer

Second law for the student, up positive: F_N − mg = ma, so a = (660 N − 588 N) ÷ 60 kg = 72 N ÷ 60 kg = 1.2 m/s² upward.

  • Answer: 1.2 m/s^2

3. At 10.0 s the scale reads 516 N. Which is true at that moment?

  1. The gravitational force on her is still 588 N
  2. The gravitational force on her has dropped to 516 N
  3. Her mass has dropped to about 53 kg
  4. The elevator is moving downward at that moment
Show the answer

The scale reads the normal force, her apparent weight. The gravitational force mg depends only on her mass and Earth's field, so it stays 588 N; the smaller reading means the acceleration points down (the elevator slows as it nears the top).

  • Correct: The gravitational force on her is still 588 N: Right: apparent weight changed, the gravitational force did not.
  • The gravitational force on her has dropped to 516 N: Earth's pull does not change in an elevator; only the scale's push does.
  • Her mass has dropped to about 53 kg: Her mass does not change during the ride.
  • The elevator is moving downward at that moment: A downward acceleration can mean slowing down while moving up, which is what happens near the top floor.

4. The cable snaps (safety brakes catch it a moment later). In the instant the elevator and student fall freely together, what does the scale read, in newtons?

Type a number in N.

Show the answer

In free fall both the student and the scale accelerate down at g. Second law for the student: mg − F_N = mg, so F_N = 0. She is weightless: the gravitational force (588 N) still acts, but nothing pushes up on her.

  • Answer: 0 N

5. Two asteroids attract each other with a force F. If the distance between their centers is tripled, the force becomes

  1. F/9
  2. F/3
  3. 3F
  4. F/6
Show the answer

F ∝ 1/r²: tripling r divides F by 3² = 9.

  • Correct: F/9: Right: the square of 3 is 9.
  • F/3: That forgets to square the factor.
  • 3F: Gravity weakens with distance; it does not grow.
  • F/6: Squaring 3 gives 9, not 6.

6. A heavy rock and a light pebble are dropped together. The rock has 10 times the mass. With air resistance negligible, which is true?

  1. The rock feels 10 times the force; both have the same acceleration
  2. The rock feels 10 times the force and falls 10 times as fast
  3. Both feel the same force and fall with the same acceleration
  4. The rock feels the same force but falls faster since it is heavier
Show the answer

Gravitational force is proportional to mass (gravitational mass), and resistance to acceleration is proportional to mass too (inertial mass). a = F/m = (10mg)/(10m) = g. Experiments find these two masses equal, which is why everything falls together.

  • Correct: The rock feels 10 times the force; both have the same acceleration: Right: 10 times the force on 10 times the inertia.
  • The rock feels 10 times the force and falls 10 times as fast: Ten times the force acts on ten times the mass, so the acceleration is the same.
  • Both feel the same force and fall with the same acceleration: The gravitational force is mg, so the rock feels the larger force.
  • The rock feels the same force but falls faster since it is heavier: The force on the rock is larger, not equal, and it does not fall faster.

Part 9 · Summary

Summary

Every two masses attract with |F_g| = Gm₁m₂/r², measured center to center; the field g = F/m is about 9.8 N/kg at Earth's surface. A scale reads the normal force, the apparent weight, which differs from mg whenever the object accelerates and is zero in free fall.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections